math.NT — Number Theory

Polynomial parametrization of quadratic Diophantine equations: decision and construction

Contributed by Bogdan Grechuk, Ashleigh Ratcliffe

We give an algorithm that, for any polynomial with integer coefficients of degree at most two, decides whether its integer zero set is a finite union of polynomial families with integer coefficients and constructs such families whenever they exist. The main step is a constructive proof that, for every nondegenerate integral quadratic form $Q$ isotropic over $\mathbb{Q}$ and every integer $m$, the integer solutions of $Q(\mathbf{x})=m$ form a finite union of polynomial families. This proves the nondegenerate case of a result announced without proof by Vaserstein. For fixed $Q$, the number of parameters in each family is bounded independently of $m$. The new descent argument treats forms in at least five variables with rational Witt index one. We also settle the previously open cases $zt-x^2-y^2=1$ and $zt-x^2-y^2=-1$ by combining Larsen--Nguyen's group parametrization with a classical reduction of binary Hermitian forms, obtaining two and three polynomial families, respectively.